This letter investigates the reconfigurable intelligent surface (RIS)-assisted multiple-input single-output (MISO) wireless system, where both half-duplex (HD) and full-duplex (FD) operating modes are considered together, for the first time in the literature. The goal is to maximize the rate by optimizing the RIS phase shifts. A novel deep reinforcement learning (DRL) algorithm is proposed to solve the formulated non-convex optimization problem. The complexity analysis and Monte Carlo simulations illustrate that the proposed DRL algorithm significantly improves the rate compared to the non-optimized scenario in both HD and FD operating modes using a single parameter setting. Besides, it significantly reduces the computational complexity of the downlink HD MISO system and improves the achievable rate with a reduced number of steps per episode compared to the conventional DRL algorithm.